Magnetic relaxation and flux creep in ceramic (Bi, Pb)-2223 HTSC

Physica C 174 ( 1991 ) 101-108 North-Holland

Magnetic relaxation and flux creep in ceramic (Bi, Pb)-2223 HTSC M . M i t t a g , R. J o b a n d M . R o s e n b e r g

lnstitut fiir Experimentalphysik VI, Ruhr-UniversitgitBochum, Postfach 10 21 48, 4630 Bochum, Germany Received 10 December 1990 Revised manuscript received 12 January 1991

The magnetic relaxation of ceramic (Bi, Pb )-2223 ( Tc= 107.2 K ) has been studied in the temperature range 3.5-100 K and in fields up to/to//= 1 T. The relaxation of the magnetization at temperatures up to 80 K can be well described by a logarithmic time decay due to thermal activation of vortex lines. The activation energy Uo obtained from the relaxation rate S= 1/M (dM/dln t) in the single barrier height model exhibits a strong temperature. Both in applied fields and in the remanent state the activation energy increases with temperature (U0=20-50 meV at T=4 K, Uo=250-400 meV at T~ 60 K). The distribution of pinning barriers has been evaluated and exhibits a peak at Uo~ 20-30 meV in the remanent state and Uo~ 15-20 meV in an applied field of/~-/= 1 T. These small values explain the rapid drop of the remanency at T> 25 K, the peak in the relaxation rate at T~ 10 K and the increase of Hot at T< 25 K.

1. Introduction Flux creep in type-II s u p e r c o n d u c t o r s due to thermal activation o f vortex lines can especially be observed by magnetic relaxation processes. Since the first report o f Miiller et al. [ 1 ] this effect was studied in detail in LaBaCuO, LaSrCuO, Y B a C u O a n d BiSrCaCuO [ 2 - 6 ]. M a n y authors f o u n d that the relaxation o f the m a g n e t i z a t i o n can be well described by the A n d e r s o n m o d e l o f thermally a c t i v a t e d fluxlines [7] which gives for the decay o f the magnetization a In t dependence:

dM ~JkuT

(1)

din t where J¢ is the critical current a n d U0 the activation energy or height o f the pinning barrier. T h e often found occurrence o f a m a x i m u m o f the decay rate d M / d l n t in its d e p e n d e n c e on t e m p e r a t u r e was exp l a i n e d by the c o m p e t i t i o n o f the linear T-term a n d the d r o p off o f the critical current with increasing t e m p e r a t u r e [6]. T h e d e p e n d e n c e on the critical current can be e l i m i n a t e d by evaluating the relaxation rate 1 dM S = M dln---t

(2)

because o f the m a g n e t i z a t i o n being p r o p o r t i o n a l to the critical current. F r o m the relaxation rate the activation energy Uo can easily be obtained. It was often found, that Uo shows a strong d e p e n d e n c e on temp e r a t u r e with Uo-~50 m e V at T---4 K a n d U o = 5 0 0 m e V up to 3 eV at T-~ 80 K. Xu et al. [ 8 ] m e a s u r e d the relaxation o f a c-axis o r i e n t e d p o w d e r specimen o f YBa2Cu307 a n d o b t a i n e d Uo-~20 m e V at low temperatures and U o - 120 m e V at T = 35 K in a field / t o l l = 1 T. They explained this t e m p e r a t u r e dependence based on the m o d e l o f Beasley et al. [ 9 ]. According to this m o d e l Uo is not the m a x i m u m depth o f the given pinning potential, but is the intersection o f the tangent aU/O(VB) at VB~-Jc a n d the U-axis (see fig. 1 in ref. [9] ). In this picture a decreasing critical current with increasing t e m p e r a t u r e leads to the increasing pinning potential o b s e r v e d in relaxa t i o n m e a s u r e m e n t s and therefore the " t r u e " pinning potential has to be higher than the m a x i m u m m e a s u r e d value. Xue et al. studied the relaxation o f melt textured a n d single crystalline YBa2Cu307 a n d o b t a i n e d a nearly linear increase o f the activation energy with increasing temperature ( Uo = 2 0 - 5 0 m e V at T-~4 K a n d U o - 3 0 0 - 6 0 0 m e V at T = 8 0 K ) [ 10,11 ]. T h e d e p e n d e n c e o f Uo on t e m p e r a t u r e has been explained in a collective pinning m o d e l with Uo=JcBVcb [7,12], where B is the magnetic induc-

0921-4534/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)

102

M. Mittag et al. /Magnetic relaxation and flux creep in ceramic (Bi, Pb)-2223HTSC

tion, Vc the pinning bundle volume and b the flux hopping distance. The increase of Vcb with T is assumed to over-compensate the decrease of J:yielding an increasing activation energy. A third explanation of the dependence of Uo on temperature was given by Hagen and Griessen [ 13 ]. They assume a distribution of pinning potentials and presented an inversion scheme to evaluate this distribution from relaxation data. Only the pinning centers with barrier depth higher than U= kTln (t/z), where r is a relaxation time, typical 10- 6 s ~ ~"~__10-12 s, are effective. So the average activation energy has to increase with T. A temperature dependent activation energy is found also by many other groups in YBa2Cu307 [ 1417 ]. In general the obtained values of Uo are about 20-50 meV at T - 4 K and U o - 0 . 5 - 2 eV at T = 5 0 80 K. For Bi containing HTSCs the obtained activation energies are of the same order of magnitude. Lessure et al. [18] obtained a temperature independent Uo---60 meV in Bi 2223 up to T/Tc~-0.5 and a rapid drop to zero at T/T¢---0.7, but they found also a temperature independent Uo -~ 150 meV in Y-123. Kumakura et al. [ 19] obtained Uo= 130 meV at T = 7 7 K and Shiet al. [20] the small value of Uo=5 meV at T = 8 K in BiSrCaCuO. It has been the aim of this investigation to find out, which of the flux creep models is applicable to our (Bi, Pb)-2223 samples and whether the relaxation rates and activation energies are comparable to the ones of other HTSCs.

reaches p = 5.2 g/cm 3 which is about 80% of the theoretical X-ray density. A detailed study of the sample preparation and the influence of the final heat treatment in air on the superconducting properties of our BiPbSrCaCuO compounds is presented in ref. [22 ]. X-ray diffraction analysis and measurements of the initial susceptibility showed, that starting from the chosen nominal composition one obtains samples with a critical temperature To= 107,2 K containing more than 90% of the 2223-phase. From electron microscopy (REM) observations the grain size can be obtained to R = I 0 - 3 - 1 0 -2 cm and R = 5 × 10 - 4 5 × 10- 3 cm for the samples with low and high density, respectively. Magnetic measurements were carried out in a sensitive vibrating sample magnetometer with a resolution of A m = 5 × 10 -5 Am2/kg in the temperature range 3.5-100 K and in fields up t o / t o l l = 1 T. The relaxation of the remanent magnetization was obtained by cooling the sample in zero field to the desired temperature and switching a magnetic field of /to//= 1 T on and immediately off. This field sweep takes about twenty seconds. Additionally the dependence of the magnetization on time in applied fields of/toll=0.1 T and 1 T was measured after cooling the samples in zero field.

3. Results and discussion 3.1. Single barrier height model

2. Experimental The samples with the

nominal

composition

Bil.84Pbo.34Sr2.o3Cal.9oCu3.osOlo+x [21] were prepared by the conventional powder method. Powders of Bi203, PbO, SrCO3, CaCO3 and CuO were mixed and decarbonized at 800 °C for 72 h and then pressed into pellets. These were heated at 855 °C for one week in pure oxygen atmosphere and afterwards at 855 °C for one week in a l l Because of the low density of our samples (p-~ 2.5-3 g/cm 3) one of them was ground again, pressed into a pellet and heated for a second time in air at 855°C. The density of this sample

The dependence of the remanent magnetization on temperature at a time tb= 50 s after switching the field off is shown in fig. 1 and exhibits two striking features: the drop of the remanency at T - 20 K and the reduction of the remanent moment of the high density sample. The decrease of MREM with increasing temperature is due to the decrease of the critical current [23 ]. The smaller value Of MREM of the high density sample may be a result of the reduction of the number of effective pinning centers yielding a lower critical current value. A more simple explanation can be obtained from Bean's model for the magnetization of type-II superconductors [24]. According to this model the critical current is given by

M. Mittag et aL / Magnetic relaxation and flax creep in ceramic (Bi, Pb)-2223 HTSC 40

103

l dM 1 S= .~/d--~ntl,= = - Uo(T)/kBT-ln(t./z)'

(5)

where tb is a typical time at which the measurement starts. The difficulties in obtaining the barrier height Up from relaxation data are described in detail in ref. [26]. For tb=50 S and 10 -6 s>_r> 10 -lz s one obtains from eq. ( 5 ):

30

.-7.

1 - ~ + 17.7<

20

10

~

i 25

- n-- _ n 50

75

100

T (K) Fig. 1. Dependence of the remanent magnetization per mass unit at tb=50 s on temperature (o, + ) low density sample; ( , ) high density material). The solid lines are cubic spline fits to the data. The rapid drop at T -~ 20 K is due to the drastic increase o f the critical current. The smaller values of the high density sample can be explained by the Bean model (see text).

Up

ka T

<-

1 - +31.5.

(6)

S

Figure 2 shows some typical relaxation measurements of the remanent magnetization. All our data can be well described with a thermal activated flux creep model. The dependence of the slope dMREM/ d l n t ( t b = 5 0 S) is shown in fig. 3(a) and exhibits a peak at T = 10 K for all samples. The smaller value for the high density sample is due to its reduced remanency. Jc can be eliminated by evaluating the relaxation rate S. According to Anderson's model S is given by S~ kBT/Up. The relaxation rate is shown in fig. 3(b). Obviously Up is temperature dependent with two temperature regions. At low temperature

25

MREM Jc ~ - - ,

R

T=4K (3)

20

where R is a typical dimension of the superconductor. If one takes R as a typical grain size the reduction of R in the high density sample leads to a smaller value of MREM with Jc being constant. A similar effect of the geometrical length scale o n remanent magnetization was reported by Yeshurun et al. [23 ]. The activation energy or barrier height Up was determined in order to explain the rapid decrease of MREM- Hagen et al. [25 ] found, that over more than 90% of the relaxation the magnetic moment is well described by:

M(t,T)=M(O,T)(1-~-~ln(l+t/z)),

~ 6K

-"'15 E <

9K

"5

10 12K 5

~ "-"-'-"-'-'-

0

(4)

where z is a relaxation time, typical 10- 6 s >-T >_ 1 0 - t 2 s. The activation energy can be obtained from the relaxation rate:

15K 20 K 25 K 3O K 5O K

. . . . . . . . . . . . . . . . . . . . . . . . . .

I0

I O0

1000

10000

t (s)

Fig. 2. Typical dependence of the remanency on time (.uoHm,,,= 1 T ) . The solid lines are ln(t)-fits. The linearity of M vs. In(t) demonstrates the validity of the thermal activated flux creep model.

104

M. Mittag et al. / Magnetic relaxation and flux creep in ceramic (Bi, Pb)-2223 HTSC 1.00

found in many HTSCs as mentioned above. The dependence of the magnetization, the slope dM/dln t(tb=50 S), the relaxation rate and activation energy in an applied field of/zoH= 1 T on temperature is shown in fig. 4. The pinning barrier Uo is at T_< 50 K approximately the same as in the remanent state. The nearly vanishing relaxation at T>--60 K leads to a diverging activation energy at this temperature. This fact corresponds to the vanishing width of the hysteresis curves at/to//> 1 T for T > 50 K [28] showing that the sample is already in the thermal equilibrium state. In fig. 5 the magnetization, the decay rate dM/dln t and the relaxation rate S in an applied field of #oH=0.1 T is plotted versus T. The relaxation rates at low temperatures are very small, because the sample is not in the critical state. So it is not meaningful to evaluate activation energies. The maximum value

o

E E 0.50

) I o.oo

A

-

- -

~

0

0.10

~vO.05

J T

0,00

0"40t ~'0.20 >

0

N +0

DNO + 0 +

It

~o.00LNi=~a~qb~m s

o

~0+

2'5

+

o

s'o

T (K)

7'5

Fig. 3. (a) Dependence of the slope dM/dln t (tb=50 s) on temperature T. (b) Temperature dependence of the relaxation rate S = l / M ( d M / d l n t ) at tb=50 S. The flux creep model yields S ~ kBT/Uo. Uo is obviously temperature dependent. The sharp peak o r s at T - 15 K can be explained by the distribution model (see text). (c) Dependence of the activation energy Uo on temperature T obtained from the relaxation rate in the single barrier height model. ( o, + ) low density; ( . ) high density sample. The solid lines are cubic spline fits to the data. These fits were used in the inversion scheme to determine the distribution of pinning barriers.

-5

"E - l O

-15

-20

0.6

r_ fill!

1 O0

0

!ooo.o ~

T (K)

2.0 0

0.06

c

1.5

"E ( T < 20 K) the slope of S versus T yields a small activation energy whereas at higher temperature (25 K < T_< 75 K) the pinning potential possesses a higher value. The relaxation rate of S-~0.025-0.05 in the temperature range 25 K_< T < 75 K is approximately the same as the "universal" decay rate S = 0 . 0 2 0 0.035 recently found by Malozemoffand Fisher [27 ] for Y - B a - C u - O superconductors. Figure 3(c) shows the dependence of Uo on temperature obtained by means of eq. (6) and exhibits a step at T-,- 25 K due to the two temperature regions of the relaxation rate. The values of Uo~ 20-50 meV at T < 2 5 K and U o - 2 5 0 - 3 5 0 meV at 25 K < T < 7 5 K are in good agreement with the activation energies

0 T (K)

k

d

O.04 >,~,°I.0

0.5

i 0.02

0.00

'

5'0

T (K)

-

I O0

0.0(

0

20

4()

60

T (K)

Fig. 4. Dependence of (a) the magnetization M, (b) slope dM/ din t, (c) relaxation rate Sand (d) activation energy Uo on temperature T in an applied field of/zoH= 1 T. ( ( o ) low density, ( . ) high density sample). The vanishing relaxation rate at T>_ 60 K leads to a diverging activation energy Uo indicating that the sample is already in thermal equilibrium. The solid lines are cubic spline fits to the data.

M. Mittag et al. I Magnetic relaxation and flax creep in ceramic (Bi, Pb)-2223 HTSC

U3(tb, T ) =

ka T i n ( 1 + tb/r)

A

~¢~--10.0

b(T)

-30,0

0.60 ,

_ d M / d M ~ -1 ln(tb/Z)=l'd--~d--~nt) . "~-

,

.

~ -

::[ Y 0 0 0 t--0

(8)

being the minimal barrier height with the capability to pin a flux-line at the temperature T; a(T) and b ( T ) describe the temperature dependence of the pinning potential U0 and the maximum possible current density, respectively. The relaxation time z can be obtained in the limit T - , 0 K (see eq. (38)) in ref. [26] ):

~-20.0

iO00

105

, .

25

. 50

. 75

1O0

T (K)

Fig. 5. Dependence of (a) the magnetization M, (b) slope dM/ din t and (c) relaxation rate S (( o ) low density, ( . ) high-density sample) on temperature Tin an applied field of/zoH= 0.1 T. At low temperatures the sample is not in the critical state, which leads to a very low relaxation rate as discussed by Griessen et al. [29] and Hagen and Griessen [30].

(9)

The values obtained for our samples by means of eq. (9) are in the range 10 - s s < r < 10 -6 s as well in the remanent state as in an applied field of/xoH= 1 T. With these values lying at the upper limit of the physically reasonable range for thermal activation one can determine the distribution of activation energies by means of eq. (7). The solid lines in figs. 3 and 4 are cubic spline fits to the experimental data and were used in the inversion scheme. Assuming an activation energy proportional to the condensation energy, we took for the temperature dependent terms a(T) and b(T) the expressions given by ref. [26]: . ~ ,

{ l + ( T I T c 2 ~ °/2

ati)=Itl,(TIT<)2)

; 1 + (T/Tc)2"~'''2 1_(T/~¢)2] •

of S - 0 . 0 5 is nearly the same as in an applied field of 1 T and again corresponds approximately to the "universal" value of Y-123 superconductors.

b(T)=(I-(T/T¢)2)

3.2. Distribution of activation energies

a(T)= l;b(T)=

The inversion scheme of Hagen and Griessen [ 13 ] was used to obtained the distribution of activation energies from the relaxation data. The distribution P(Uo) is defined in such a way that P(Uo)/dUo is the fraction of barriers with activation energies between Uo and Uo+dUo at T = 0 K. The distribution is then given by

The distribution functions P(Uo) for our samples in the remanent state are shown in fig. 6. The peak around 20-30 meV is similar to the one at 60 meV found by Hagen and Griessen for the polycrystalline YBa2Cu307-sample studied by Tuominen et al. (see fig, 21 in ref. [26] ) and the 50 meV obtained in ceramic T1-1223 sample by our group [31 ]. A slightly different result is presented by Xue et al. [ 10 ] with a monotonically decreasing distribution function in Y-123. By means of the distribution shown in fig. 6 one can explain the rapid drop of the remanency at T > 25 K in our samples. The lowest barrier height needed to pin a flux-line at temperature Tcan be ob-

P(U~(tb, T) ) d(a(T)

dM'~/(b(T) d \ T ~

=~\~d-~-tnt] with

T b(T)

) (7)

With the same exponents as in ref. [26] ( n = 2 and m = 0) one has simply

1- ( T/T~) 4 .

M. Mittag et al. /Magnetic relaxation and flux creep in ceramic (Bi, Pb)-2223 HTSC

106

mental data, by integrating the distribution function

0.05

[261

T=20K ~ H m a x = 1 TI

0.04

= 30K

0.03

(Uo(T)) =

~Ui(tb,r) P( Uo)d Uo ~-u~ (P(Uo)/Uo)dUo

(lo)

~(tb,T) o.. 0 . 0 2

c 0.01

0.00

0.01

0.1 Uo ( e V )

Fig. 6. Distribution of activation energies for all samples in the remanent state. The arrows indicate the thermal energy of the flux-lines at the given temperature with In (tb/T) = 20 ( z = 10 - 7 s). At T > 30 K only the few pinning centers with barrier heights lying in the tail of the distribution are effective, leading to a low critical current at high temperatures (see fig. 1 ). ( ( a, b ) low density; (c) high density sample).

rained by means of eq. (8). At T ~ 20 K this temperature corresponds approximately to the maxim u m of the distribution; at T ~ 30 K only the few pinning centers with barrier heights lying in the tail of the distribution are effective, leading to a drastic decrease of the remanency at this temperature. The distribution picture yields also a possible explanation for the decrease of the first critical field Hc~ at low temperatures (T-< 25 K) obtained in our samples from the virgin magnetization curves by the kink point method [28]. Flux lines are not able to penetrate the sample at low temperatures because the vortex lines owing to their small thermal energies are pinned at the surface. The peak of the relaxation rate S at T - l 0 K is also due to the relatively small value of most of the barrier heights leading to a loss of the capability to pin vortex lines. The result is a reduction of the relaxation at higher temperatures. The increase of S at very low temperatures up to its peak value is due to the increasing thermal energy of the flux-lines. One can obtain the average effective activation energy ( U o ( T ) ) , which should be equivalent to the pinning potential determined by fitting a single activation energy expression (eq. ( 4 ) ) to the experi-

The solid line in fig. 8(a) is obtained by numerical integration of the distribution in fig. 6 and shows a quantitatively good agreement with the activation energies determined in the single barrier height model. The step at T -~ 25 K corresponCls to the sharp drop of the distribution function at Uo > 40 meV. The distribution of activation energies in an applied field ofHoH= 1 T is shown in fig. 7 and exhibit a peak at lower barrier heights ( Uom~ ~ 15-20 meV). This may be due to the field dependence of the activation energy, which is often obtained by fitting an Arrhenius term to resistivity data describing the energy dissipation by flux line motion [32,33 ]. The even smaller barrier heights in an applied field lead to an approximately linear dependence of the single barrier activation energies on temperature (see fig. 8 ( b ) ) at T_< 40 K. The diverging activation energy at T>_ 50 K has already been explained above.

0.05

0.04

~" 0.03

a. 0 . 0 2

0.01

0.00 0.01

•

,

. . . . . . .

0.1

uo (ev)

Fig. 7. Distribution of barrier heights in an applied field of Moll= 1 T. The m a x i m u m of P(Uo) is at a lower energy than in the remanent state due to the field dependence of the pinning potential. (a) low density; (b) high density sample).

M. Mittag et al. / Magnetic relaxation and flux creep in ceramic (Bi, Pb)o2223 HTSC 0.30 o

0.20

0.10

0.00

0

20

40

60

80

1O0

T (K)

0.40

0.20

0.00

N

0

10

20

30

40

50

T (K)

Fig. 8. (a) Dependence of the activation energy of a low density sample in the remanent state obtained from the relaxation rate in the single barrier height model ( + ) . (b) Uo(T) of the high density sample in an applied field of/to//= 1 T. The solid lines are the effective activation energies ( Uo(T) > obtained by means of eq. (10) by integrating the distribution function. This effective pinning energy is in good agreement with the values obtained in the single barrier height model.

4. Conclusion The magnetic relaxation of the polycrystalline (Bi, Pb)-2223 HTSC has been investigated. The remanent magnetization is reduced in a high density sample, which may be according to Bean's model of the magnetization of type-II superconductors a result of the relatively small grain size compared to the low density material. In all samples the remanency exhibits a rapid drop at T>_ 25 K due to the drastic decrease of the critical current at this temperature. All relaxation data can be well described by a logarithmic time decay indicating the validity of the flux creep model due to the thermal activated vortex lines. The activation energies were determined in the single barrier height model from the relaxation rate S= 1/M (dM/dln t) (tb= 50 S). S lies between 0.02 and 0.05 in the temperature range 25 K < T_<60 K, in good agreement with the "universal" value of

107

S=0.02-0.035 found in Y - B a - C u - O superconductors [27]. As well in the remanent state as in an applied field of/~oH= 1 T the activation energies Uo are increasing with T. The values of Uo=20-50 meV at T < 2 0 K and Uo< 250-350 meV at 30 K < T < 75 K in the remanent state are in good agreement with pinning potentials often found in HTSC. The vanishing relaxation at T > 60 K in an applied field o f / t o / / = 1 T yields a diverging activation energy indicating that the sample is already in thermal equilibrium. The distribution of pinning barriers determined with the inversion scheme of Hagen and Griessen [ 13 ] peaks at Uo=20-30 meV and Uo-~ 15-20 meV in the remanent state and in/zoH= 1 T, respectively. These values are slightly smaller than Uomax ~---60 meV found in Y-123 [26] and Uomax=50 meV in T1-1223 [31 ]. These small values lead to the drastic decrease of the remanency at T > 2 5 K, the peak in the relaxation rate at T = 10 K and the increase of He, at T < 2 5 K. In this model it is also possible to explain quantitatively the dependence of the single barrier activation energy Uo on temperature. The integration of the distribution function leads by means of eq. (10) to an effective pinning potential ( Uo(T) >, which is in good agreement with the activation energy Uo(T) obtained in the single barrier height model. So we have found that a distribution of activation energies describes our relaxation measurements in a quite satisfactory manner. For this reason we consider the distribution model as a very useful tool for the analysis of magnetic relaxation processes in highTc superconductors.

Acknowledgement The financial support of the Bundesministerium f'fir Forschung und Technologie (project 13 N 5713 ) is gratefully acknowledged.

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M. Mittag et al. / Magnetic relaxation and flux creep .in ceramic (BL Pb)-2223 HTSC

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